Journal of Mathematical Biology

The dynamics of gene regulatory networks are governed by intrinsic noise, stemming from the random nature of biochemical reactions, and by extrinsic noise, arising from fluctuations in cellular components and environmental conditions. Together, these sources can compromise the reliability of predictive computational models if not properly accounted for, and capturing both effects within a single framework remains a non-trivial task in computational biology. In this work, we propose an uncertainty quantification framework that addresses these two contributions jointly: intrinsic stochasticity is described through a partial integro-differential equation (PIDE) for the protein probability density function, whereas extrinsic noise is represented as parametric uncertainty in the kinetic parameters. The propagation of the uncertainty is carried out via an intrusive polynomial chaos expansion (PCE), in which the PCE coefficients are obtained from a stochastic Galerkin projection of the PIDE, yielding a coupled deterministic system that is solved with standard numerical methods. We illustrate the approach on a positive autoregulatory gene network with one and two uncertain kinetic parameters. The proposed approach accurately reproduces the mean, variance, and full protein probability density function, including the bimodal distributions, at a substantially lower computational cost.

The study of tissue dynamics has been stimulated during the last decades thanks to the use of quantitative descriptions, with the development of several theoretical and computational frameworks, many of them revolving around the notion of reaction-diffusion systems, eventually with additional structure variables beyond time and space. The use of structure variables can accommodate phenotypic traits. In this work, we study a family of competition models, where a given population depends on a resource (e.g. oxygen) and several populations are competing for it. Our quantitative description incorporates phenotypic traits and heterogeneity at the level of cell cycle variations, which influence replication rates via oxygen consumption. This enables us to replicate the fitness of specific subpopulations to environmental conditions (e.g. oxygen shortage or external influences). Using numerical simulations, we show that such models display dynamical pattern formation in the form of coupled travelling wave profiles that expand or retreat at the same wave speed. The full theoretical analysis of such dynamics is quite involved; to circumvent this difficulty, we introduce a quasi-stationary approximation for the resource dynamics. We find that this approximation can reproduce the overall behaviour very accurately, with the additional benefit of allowing theoretical treatment of the reduced model. In this way, we provide estimates on the wave speed which are numerically shown to be robust across a wide range of macroscopic parameters of the full model. The wave speeds are thus found to depend strongly on the proliferation rate of the fittest population, resembling a winner-takes-all dynamics.

Spatial population structure and seasonality are both central to the spread of many infectious diseases of plants, animals and humans. While seasonal forcing in transmission often plays an important role in epidemiological models of a wide range of infectious disease, and we now have some theoretical understanding of the dynamical impacts of spatial structure, the combined effects of these two ubiquitous processes has not been examined in detail. Here, we develop a novel model to explore the combined influence of spatial structure and temporal variability on disease dynamics. Spatial structure is represented using a lattice-based approach with near-neighbour interactions, while temporal variability is included through regular, seasonal, variation of the transmission rate. We use bifurcation analysis of a pair approximation of the full spatial model to identify the parameter regimes associated with qualitatively distinct dynamical behaviours. The model exhibits a remarkably wide range of complex dynamics, including limit cycles, quasi-periodic cycles, multi-year cycles, chaotic dynamics and bistability between these different states. In particular, complex dynamics occur when reproduction is predominantly local, with the dynamics depending critically on the amplitude of the seasonal transmission rate. We show how high transmission rates, high birth rates and in particular low recovery rates are requirements for complex dynamics. We predict that SI-type disease interactions in plant pathogen systems will show complex dynamics even with relatively global transmission dynamics.

A novel mathematical framework to define the threshold of action potentials in excitable cells is presented. Unlike previously applied methods that rely on approximations or bifurcations, the approach focuses on the geometry of membrane potential trajectories. The changes in concavity during the upstroke of an action potential can be directly obtained from a time series of voltages. The concavity criterion is then extended to models based on autonomous dynamical systems where the changes in concavity can be obtained analytically from a curve of inflection points in phase space. The inflection point manifold defines a region required for excitability: all the orbits that cross it contain action potentials, and all the trajectories that contain action potentials are in it. This analytical principle can then be used to define excitability in a dynamical system, and also a measure of excitability that enables quantification and comparisons of excitability across dynamical system. The measure provides a way to compare the excitabilities of systems that model neurons with different electrophysiological phenotypes and consider different stimulus conditions. The traditionally vague physiological concept of electrical excitability is transformed into a rigorous analytical description by considering the time-dependent curvature of the membrane potential. The criterion is robust across smooth, single compartment models of electrical excitability and can be can be extended to single compartment models in higher dimensions, and multicompartment models as well.

Tumors consist of heterogeneous phenotypic cells, such as normoxic cells, which are highly proliferative, and hypoxic cells, which are less proliferative. Their phenotypic switching depends on tumor microenvironmental factors, such as oxygen and nutrient concentrations supplied by local blood vessels. However, during ongoing angiogenesis, the process of sprouting new blood vessels at the tumor site from pre-existing blood vessels, and how this phenotypic switching affects and impacts tumor growth, remains poorly understood. In this article, we formulate a mathematical model to elucidate the crosstalk between vasculature and tumor cellular heterogeneity during tumor progression. The model results show a strong agreement with the experimental data. Our simulation results demonstrate that ongoing angiogenesis increases tumor growth rate. In addition, we observe that the influence of hypoxic cells on phenotypic switching from normoxic to hypoxic is more pronounced than their influence on the transition from hypoxic to normoxic. Furthermore, we perform a global sensitivity analysis using the Sobol's method to assess the importance of the model's parameters. It highlights that the volume at which blood vessels attain half-maximal rate has the maximum effect on the model.

Can observations distinguish a bloom supplied from within a study volume from one supplied across its boundary? We develop a theoretical framework for that question at plankton bloom onset, conditional on a predeclared, observed or calibrated onset event and a declared set of environmental paths, biological responses, and model forms. The estimand follows source-event labels through forcing-dependent survival and genotype-specific growth. Its central certificate asks whether the local onset fraction is invariant over every source history that produces the same time-expanded observation record. For polyhedral history fibers, a Charnes-Cooper transformation computes both sharp dynamic-data endpoints as linear programs. When each source instead has a fixed normalized onset signature, the certificate reduces to a row-space test; uncertain signatures require a joint lifted program. For a finite compatible scenario ensemble, admissible fractions are the union across scenarios, and a point is justified only when every nonempty scenario gives the same singleton. A synthetic two-genotype witness gives the same observed total but local fractions of 2/3 and 1/3 under reversed forcing-response gains. The observer, mixture, and optimization ingredients are established; the contribution is their target-specific synthesis around source at onset. The framework is diagnostic rather than predictive. It specifies what a study must measure--local sources, boundary inflow, forcing, response, timing, and carrier signatures on one declared window--and returns an interval when missing components have justified bounds, including [0, 1] when they remain unconstrained.

Zero-dimensional chemical master equations, ordinary differential equations, and compartmental population models replace spatial stochastic biological systems by vectors of total counts or densities. This study asks when that projection is exact and whether information retained in spatial correlations can diagnose its practical failure. Exact Markov closure is characterized by an aggregate-rate lumpability condition: for every retained transition, the sum of microscopic transition rates must be constant over all spatial configurations with the same counts. Violations are connected to BBGKY-type correlation hierarchies and to mean-field, pair, and triplet closures. Conditional rate, finite-time predictive, memory, path-space, and correlation Kullback-Leibler risks quantify distinct losses. An exactly solvable two-compartment reaction separates structural non-closure from recovery of a well-mixed law under fast hidden mixing. Copy number and a spatial mixing-interaction ratio connect concentration, volume, diffusion, and reaction parameters to practical screening, including an Escherichia coli-scale example. The same projection logic is evaluated in controlled spatial susceptible-infectious-removed and predator-prey benchmarks. Across mixed and segregated initial conditions and four mobility regimes, pair-correlation risk was strongly associated with the error of the corresponding zero-dimensional ordinary differential equations (Spearman correlations 0.95 and 1.00; pooled 0.99). A nearest-neighbour exchange sensitivity analysis preserved the positive risk-error ranking. These benchmarks do not establish a universal threshold, but support correlation information as a transferable diagnostic for selecting among count, pair, higher-order, and explicit spatial descriptions.

Within the studies of stochastic gene expression, apart from the variability of copy number of gene products, the problems of threshold crossing of those products are biologically important as they often lead to terminal cellular events. Here, we study the threshold crossing problem of the messenger ribonucleic acid (mRNA) and present an exact probability distribution of first passage times in Laplace space. The function furnishes moments of any order and also predicts the characteristic time of the exponential tail of the distribution, which we match against Gillespie simulations. We find that all the measures of relative fluctuations of the threshold crossing times show U-shapes within this simple model of mRNA, as was found earlier in more mathematically involved models of threshold crossing time statistics of proteins. Furthermore, we extend the exact formula to include the phenomenon of DNA duplication and the corresponding doubling of transcription rate. As expected, the distribution varies considerably depending on the onset of the duplication stage within the cell cycle.

Structural network representations of metastatic dissemination typically focus on static topology without resolving transport dynamics, relaxation timescales, or steady-state behaviour. Here, we formulate a discrete Markovian transport model on a directed higher-order network with transition rates derived from qualitative clinical affinity classes. By constructing a non-Hermitian row-stochastic transfer operator, we characterise the relaxation dynamics through its spectral decomposition. The system exhibits a fast-mixing regime characterised by a spectral gap of {gamma} {approx} 0.67, corresponding to a characteristic relaxation timescale of {tau} {approx} 1.49 discrete steps, with the influence of the primary tumour origin progressively attenuated during dissemination. Convergence towards a non-equilibrium steady state (NESS) is accompanied by a reduction in Shannon entropy, concentrating probability mass within specific topological sinks. This spectral relaxation delineates two distinct dynamical regimes: early transient dissemination (n < {tau}), dominated by local organ-specific transition probabilities (organotropism), and the asymptotic regime (n > {tau}), determined increasingly by the global transport architecture of the network. Comparison with independent clinical and autopsy observations across 21 primary tumours and 23 target organs indicates that the predicted stationary distribution is consistent with the observed hierarchy of metastatic organ involvement.

With 302 neurons and a rigorously characterized connectome, the nematode Caenorhabditis elegans represents a powerful model organism to study the fundamental roles of neuronal circuits in behavior. However, despite the breadth of research, many questions remain unanswered regarding how these organisms are able to successfully navigate their environment. Here, we present a biologically grounded dynamical circuit model for the investigation of sensory-guided behavior during C. elegans chemotaxis. Our mathematical model consists of the chemosensory neuron AWA, interneurons RIM and RIA, motor neurons, including SMDs and RMDs, and body wall muscles that provide proprioceptive feedback through stretch receptors. After optimization with an evolutionary algorithm, the model locomotes effectively toward a chemical attractant, realistically capturing nematode chemotactic behavior. Chemotaxis is ensured by sharp turns, which resemble the omega turns of living nematodes, as a key emergent property of the model. The sharp turning behavior is triggered by decreases in the concentration of the attractant. These result in reduced AWA activity, which in turn triggers disinhibition of RIM and subsequent changes in RIA oscillations. The ensuing coordinated changes in downstream motor neurons activity patterns produce sharp turns, which correct the nematodes path, so that the model worm heads toward the attractant, and remains at its proximity, after it reaches the gradient peak. The proposed framework, along with its emergent dynamics, provides new insights into the minimum requirements for C. elegans circuitry to display major features of its chemotactic behavior, including omega turns. In parallel, it generates experimentally testable hypotheses with respect to the participating neuronal elements.

Animals use long-range signals, such as hormones and neural signals, to coordinate the actions of distant organs. There is no precise, quantitative framework that explains the problems these control systems must solve and thus predicts their behavior under varied conditions. We consider this problem in the context of blood glucose regulation by the hormone insulin, the failure of which produces diabetes. We show that existing mathematical models of glucose regulation admit equivalent control strategies with no hormones at all, and thus cannot explain the need for hormonal regulation. We therefore introduce a minimal model of inter-organ variations in local glucose, and show that control strategies based on local glucose measurements face severe trade-offs between different control objectives. In contrast, we show that hormonal control signals from the pancreas can overcome these limitations. By exposing the benefits of hormonal control, our work paves the way to a detailed understanding of physiological design principles, with possible implications for the engineering of an artificial pancreas.

Classical neurogeometric models describe the primary visual cortex as a fibered structure in which retinal position and local orientation are coupled through the geometry of the roto-translation group. We extend this approach to the visuomotor cortex by modeling it as an assemblage of visual and motor cortical geometries. The model combines orientation-selective representations, analogous to those of the primary visual cortex, with movement-direction-selective representations, analogous to those of the primary motor cortex, in order to describe the mixed visual and motor selectivity observed in the visuomotor cortex. We introduce a coupled visuomotor structure in which visual orientation and motor direction coexist over a common spatial plane and interact through a relative-orientation constraint. Neural responses are modeled by orientation- and direction-dependent profile functions, and preference maps are obtained from vectorized population responses. Numerical simulations generate visual, motor, and mixed visuomotor response maps. A competition rule between visual and motor responses produces incidence ratios close to experimental observations in macaque visuomotor cortex. This framework provides a first neurogeometric approximation of visuomotor functional architecture and a mathematical setting for studying visually guided action.

A fundamental biological process of transcription occurs in the cell nucleus, which is a complex medium that also contains multiple heterogeneous structures known as biomolecular condensates. Interestingly, some of these condensates contain mRNA molecules in addition to proteins, suggesting an important cellular role in transcription that is not yet well understood. In this work, we develop a minimal theoretical framework for quantitative investigation of the role of reversible mRNA condensation in transcription. Our discrete-state stochastic approach accounts for the most relevant processes, allowing us to explicitly evaluate the properties of the system and clarify the effects of condensation. Analytical calculations supported by computer simulations suggest that reversible mRNA condensation influences the transcription processes by maintaining a constant level of free mRNA in the nucleoplasm while lowering the degree of stochastic noise and increasing the robustness against external perturbations. Physicochemical arguments are presented to explain these observations. The proposed theoretical framework elucidates important microscopic aspects of transcription, providing a convenient quantitative tool for investigating complex biological phenomena.

BackgroundPlatelet activation in flowing blood is a multiscale process in which vessel-scale hemodynamics, red blood cell (RBC) mechanics, adhesive receptor interactions, and intracellular signalling jointly determine thrombotic risk. Individual components are well studied, but a single reduced description that carries each explicitly from vessel-scale flow to mechanosensitive calcium entry, with dimensionally consistent couplings, remains uncommon. ObjectivesWe develop and analyse a reduced, six-module mechanobiological framework for platelet priming spanning the cascade from hemodynamic shear to mechanosensitive calcium entry, and we delineate which elements are supported by existing evidence and which are new, testable hypotheses. MethodsThe framework comprises six coupled modules: (I) hemodynamic forcing from the incompressible Navier-Stokes equations, with an objective principal-strain-rate measure for extensional flow; (II) RBC-mediated platelet margination and near-wall delivery, closed by a near-wall arrival flux; (III) von Willebrand factor (VWF) activation with a bounded kernel and glycoprotein Ib (GPIb) catch-slip capture, resolved through an explicit contact area and a bond-dependent mobility that progressively immobilises wall-interacting platelets; (IV) a single-load membrane-stimulus formulation; (V) mechanosensitive gating and a dimensionally consistent cytosol-store calcium model with extracellular influx; and (VI) a phenomenological mechanical-memory state. We formally derive that the single-platelet stochastic dynamics and the continuum population balance form a Fokker-Planck pair, with the spatially varying diffusivity handled by an explicit drift correction. ResultsThe framework yields a family of mechanochemical dimensionless groups delineating priming regimes. Its central prediction is reformulated as a falsifiable, history-sensitive signature: in a conditioning-test protocol, a low-tension conditioning block charges the memory state, and a fixed sub-threshold test pulse then reports a delay-dependent calcium facilitation that decays on the memory time{tau} m and is distinguishable from no-memory gating, channel adaptation, and residual-calcium priming. We show explicitly that the previously proposed pulsatile-versus-monotone contrast is a nonlinear convexity/thresholding effect of the gating nonlinearity--its difference-in-differences is approximately zero-- and is therefore not a valid test of memory; the conditioning-test signature is. A second prediction links RBC stiffening to reduced near-wall delivery and captured-platelet calcium response, upstream of intrinsic platelet signalling. ConclusionsThe framework provides a dimensionally consistent, mechanistically grounded and hypothesis-generating description linking hemodynamic forcing to mechanosensitive calcium entry. It demonstrates how history-dependent platelet priming may arise from a phenomenological sensitisation state and proposes a conditioning-test protocol for comparison against adhesive, channel and intracellular-store persistence. The framework is calibratable rather than validated, and the quantitative outputs shown use representative uncalibrated parameters.

After introduction, many non-native marine species are dispersed planktonically. Secondary spread within the non-native range has been shown to prevent the establishment of the introduced species if the advection of larvae prevents sufficient return of larvae to maintain the population in the face of competition with native species. However, those studies have largely neglected the effects of spatial variation in alongshore larval transport. We examine the introduction of a novel species with planktonic dispersal into a more realistic coastal environment which includes spatial variation in larval transport estimated from the Mercator Ocean 1/12th degree global circulation model. The introduction may either be from a distant habitat, or through range expansion. We find that there are locations in the global coastal ocean where introduced species are more likely to persist because of spatial variation of coastal currents. These include regions where alongshore larval transport diverges, such as estuaries. The location where a non-native species is introduced may not be where it flourishes - it cannot be assumed that the region where invading species are first noticed to be abundant is the region where it was introduced. We extend closed-population theory to open coastal systems to estimate persistence as a function of local circulation, habitat extent, and the competitive advantage of the introduced species. Software is provided which allows the estimations of regions where introduced species are more likely to persist and flourish as a function of larval depth behavior, planktonic duration and release timing.

Cross-feeding relationships shape the composition of many microbial communities, yet the evolutionary processes that give rise to them remain poorly understood. Most theoretical and experimental work has therefore focused on minimal scenarios, particularly the stable cross-feeding polymorphisms that evolve in asexual populations growing on a single energy source (Helling et al., 1987). Yet replicate experiments do not always produce cross-feeding populations, raising the question of why genetically identical populations evolving under identical conditions can follow different evolutionary trajectories (Treves et al., 1998). Here we present a bare-bones agent-based model of evolution in a chemostat. We show that selection for energy acquisition alone is sufficient to promote the evolution of cross-feeding, without invoking mechanisms specific to metabolic exchange. The resulting communities nevertheless differ across replicate simulations, reproducing the qualitative variability observed experimentally. Significance StatementMicrobial communities often depend on cross-feeding, in which one cells metabolic product becomes anothers energy source. Existing explanations typically invoke trade-offs between metabolic tasks or other mechanisms specific to cross-feeding itself. Using large-scale in silico simulations of evolution in a chemostat, we show that no such explanation is required. A population that competes for metabolic energy by utilizing a primary resource and then releasing a product that may itself serve as an energy source can evolve into a mixed population of organisms that specialize in the primary resource alongside others that specialize in the secondary one. Energy-based probabilistic death and reproduction are sufficient to produce this coexistence and to reproduce the mixed outcomes seen in laboratory evolution experiments.

Tumor spheroid mechanics arise from both the mechanical properties of individual cells and the adhesive interactions that organize them into tissues. The relative contribution of these two factors to the bulk mechanical behavior, however, remains difficult to disentangle experimentally. Here, we develop a computational model of micropipette aspiration to compare the mechanical response of isolated cells and multicellular spheroids within a common computational framework. By independently varying single-cell stiffness and cell-cell adhesion, we quantify their effects on aspiration dynamics, effective elastic modulus, and viscoelastic relaxation. Our results show that increasing single-cell stiffness substantially alters the mechanics of isolated cells but has limited influence on the effective elastic modulus of multicellular spheroids. In contrast, changes in cell-cell adhesion produce pronounced effects on spheroid effective elastic modulus. Nevertheless, both parameters increase the retardation time governing the transition from the initial elastic response to long-time viscous deformation. These findings suggest that multicellular elasticity is governed primarily by intercellular mechanical coupling, whereas the dynamical response to applied stress depends jointly on cell-scale mechanics and cell-cell adhesion.

Maintaining a prescribed composition in engineered microbial consortia is difficult because small fitness differences can drive competitive exclusion. We study a two-strain consortium in continuous culture and develop a feedback architecture that regulates composition by selectively slowing the fast strain as a function of the population ratio. At the population level, we derive an idealized ratio-feedback law with a tunable positive coexistence equilibrium. We then propose a biomolecular realization using orthogonal quorum sensing, an sRNA-based ratiometric controller, and a ppGpp-mediated growth actuator. Exploiting the separation between slow population growth and faster intracellular controller dynamics, we use singular perturbation theory to show that, for sufficiently fast controller dynamics, the full implementation model inherits the coexistence equilibrium and its local stability properties from the reduced model. Numerical simulations validate the reduction and show how weaker timescale separation or loss of the assumed molecular regime degrades performance.

Time compaction theory is a general framework explaining how a brain can efficiently deal with dynamic situations occurring in, e.g., sports games. It involves a geometric representation of the time dimension, which enables effective learning and strategic action planning. The theory has recently received experimental support in humans. However, its current computational model has an important limitation: it does not account for deliberate waiting and speed modulation, behaviors ubiquitous in natural environments. This work substantially extends the original model formulation by a dimensional lifting of an n-D workspace into (n + 1)-D mental space, where time remains geometrically embedded. The proposed biologically inspired computational model can generate adaptive behavior across increasingly complex situations, from navigation in everyday social environments to competitive sports. Furthermore, by actively conditioning the expected responses of other agents and stabilizing future predictions, we introduce the concept of uncertainty points in sequences of generalized cognitive maps to support the generation of adaptive strategies in interactive environments, where future prediction has a limited time horizon. Thus, we provide a mechanism for chaining short-term solutions into long-term strategies, which is illustrated by simulating the behavior of a player in a real football game. Author summaryHumans often anticipate future interactions in dynamic environments. Many behaviors, such as avoiding other pedestrians, letting someone pass through a narrow corridor, or reproducing the kind of dribbling maneuvers performed by elite football players, require deciding not only where to move but also when to move. Existing theories suggest that the brain simplifies such situations by representing future interactions as static spatial maps, making them easier to learn and recall. However, current computational models cannot naturally account for common behaviors such as waiting, slowing down, or modulating speed. Here we show that these behaviors readily emerge if the model space is extended by an additional virtual coordinate that encodes accumulated waiting rather than physical time. The proposed model simultaneously admits a wide variety of behaviors, including speed modulation, multigoal decisions, and compound actions, while preserving the principles of time compaction. We illustrate the model in everyday situations and by reproducing two real football plays, comparing the observed behaviors with model simulations. Our results suggest computational principles through which the human brain may efficiently represent, memorize, and exploit dynamic situations.

Microbial populations generally evolve in fluctuating environments under time-varying conditions. These are often described by binary switching models, sometimes seen as coarse-grained feast-famine cycles, in which resource availability switches abruptly between abundant and scarce conditions. However, experimental studies suggest that feast-famine environments actually exhibit more complex temporal dynamics. Here, we study how two strains, one growing slightly slower than the other, compete for the same resources in fluctuating environments comprising a finite number of intermediate states, each having its own carrying capacity. Environmental switching between these states and their carrying capacities represents gradual changes in nutrient availability. This class of multi-state stochastic switching models can be interpreted as a coarse-grained description of feast-famine cycles and allows us to investigate strain competition under the gradual recovery and depletion of resources. By computational and analytical means, we characterise the population dynamics in these multi-state fluctuating environments. In particular, we study how the switching rates and distribution of carrying capacities affect the population-size statistics, fixation probability, and mean fixation time. By comparing these results with their counterparts in binary environments, we clarify how the frequency and amplitude of environmental fluctuations influence population dynamics in coarse-grained feast-famine cycles.